S. Casciati
Department DARC, University of Catania, Italy
criticism usually raised against these techniques is related to their computational effort, but the DE procedure originally proposed by Storn & Price (1997) significantly shortens the implementation runtime with respect to the common genetic algo-rithms, by making use of simple sum and differ-ence operations to perform the cross-over and the mutation tasks. In Casciati (2008), it is shown that the convergence can be reached within a reasonable number of iterations even when the DE procedure is applied to high dimensional problems. The main advantages of this procedure include its easiness of implementation, its readiness of adaptability for the solution of different optimization problems, and its high rate of convergence. In this paper, an optimization algorithm of differential evolution type is employed for the reliability assessment in RBDO problems.
Several approaches were proposed in literature to tackle the RBDO problem. A recent overview of the subject can be found in Aoues & Chateauneuf (2010), where a benchmark study is also developed to enable a comparison of the numerical perform-ance and accuracy of the different RBDO meth-ods. The benchmark study consists of a set of examples using mathematical and finite element models. Its aim is to define the validity domain of each method and to lead to the choice of the most suitable approach depending on the engineer-ing structure to be solved. In the present work, it is used to prove that the validity domain of the one-level RBDO method based on the Karush-Kuhn-Tucker (KKT) conditions of the first order reliability method (Rackwitz 2002) can be signifi-cantly enlarged by exploiting the differential evolu-tion technique as soluevolu-tion strategy. Indeed, when the RBDO problem is characterized by a strong nonlinearity of the limit state function, an unsat-isfactory performance of the KKT approach is achieved by adopting a traditional optimization framework based on a Sequential Quadratic Pro-gramming (SQP) algorithm. Indeed, a high sensi-tivity to the initial point leads to an initial increase of the objective function when the algorithm starts far away from the optimal point. Furthermore, a numerical instability of the optimization algorithm and a consequent lack of convergence occur as the The methods of risk and reliability analysis in
civil engineering applications mainly developed during the last three decades and they are increas-ingly gaining importance as decision support tools (Faber & Stewart 2003). In particular, the Reliability-Based Design Optimization (RBDO) methods drive the selection of the design param-eters by seeking the best compromise between cost and safety. When the non-deterministic nature of the input data is explicitly considered in the struc-tural optimization problem, the adoption of suit-able algorithms in order to evaluate the reliability constraints is required. The structural reliability assessment can be pursued either by stochastic simulations or by moment methods. Since the Monte Carlo simulations are computationally expensive, the moment methods, such as the first and second order reliability methods (FORM and SORM), are usually preferred due to their simplic-ity and efficiency. Nevertheless, their computa-tional cost is still high with respect to deterministic optimization and it represents the main obstacle to the application of RBDO methods to practical engineering structures. The original formulations of these traditional approaches for structural reli-ability calculations can be conveniently revisited in light of the most recently emerging soft computing techniques in order to overcome their limitations and to improve their performance.
Differential Evolution (DE) algorithms are heu-ristic methods for the direct search of the global optimum of an objective function. With respect to the classical gradient based algorithms, they do not require any assumption on the differentiability of the objective function and they lead to results which are independent of the initial guess. The null gradient condition is replaced by a convergence criterion which is satisfied when the optimal points retrieved from two successive iterations fall very close to each other in the solution space. There-fore, such a family of algorithms is particularly suitable to solve optimization problems which are characterized by strong nonlinearities. In general, the adoption of an evolutionary strategy to solve an optimization problem allows more freedom in the selection of the objective function formulation, whose gradient has not to be computed. The main
ICASP Book I.indb 134
ICASP Book I.indb 134 6/22/2011 12:42:00 AM6/22/2011 12:42:00 AM
nonlinearity of the limit state function increases.
These issues suggest to resort to evolutionary techniques which are, in general, capable to reach the optimal solution with simplicity in theory and easiness of programming, even when a highly nonlinear and partly non-differentiable objective function which exhibits many local minima is con-sidered. Whereas the gradient methods operate on a single potential solution and look for some improvements in its neighborhood, the global opti-mization techniques, represented herein by the so called evolutionary methods, maintain large sets (populations) of potential solutions and apply to them some recombination and selection opera-tors. In particular, the DE approach is adopted as a robust statistical, parallel direct search method for cost function minimization. Besides its good convergence properties, the attractiveness of the method also consists of requiring only few input parameters which remain fixed throughout the entire optimization procedure. The influence of the initial guess on the results is removed by apply-ing the procedure to a population of NP equally important (N + D)-dimensional vectors, instead of referring only to a single nominal parameter vec-tor. At the start of the procedure (i.e., generation P = 0), the population of vectors is randomly cho-sen from an uniform probability distribution. For the following (P + 1) generation, new vectors are generated according to a mutation scheme. With respect to other evolutionary strategies, the muta-tion is not performed based on some separately defined Probability Density Functions (PDF), but it is solely derived from the positional infor-mation of three randomly chosen distinct vectors in the current population. The trial parameter vector is built by adding the weighted difference between two randomly chosen population vectors to a third random vector. This scheme provides for automatic self-adaptation and eliminates the need to adapt the standard deviations of a PDF.
As it is commonly done in the evolutionary strat-egies, a discrete recombination or cross-over is then introduced to increase the diversity of the new parameter vectors. The latter ones are even-tually compared with the members of the current population and they are chosen as their replace-ments in the next generation only when they lead to an improvement of the objective function value toward the search direction.
The formulation of a pertinent objective func-tion is crucial to the design process and the mul-tiple objectives have to be adequately weighted.
Such an approach can be naturally extended to include also a robustness requirement as discussed in (Casciati 2007) and (Casciati & Faravelli 2008).
The differential evolution strategy is applied to solve some of the mathematical problems proposed by the benchmark study. In particular, the efficacy of the method is discussed with reference to prob-lems characterized by cost functions with different degrees of nonlinearities and problems character-ized by multiple limit states. The results show that the DE strategy is quite reliable in terms of find-ing the solution. Nevertheless, several counterac-tions need to be performed along the optimization process in order to avoid local minima and to limit the computational burden. Two further improve-ments, such as the adoption of response surface approximations for the performance functions to avoid large finite element computations and the fragmentation of the design space in sub-domains to decrease the number iterations to convergence, are considered as possible themes for future work.
REFERENCES
Aoues, Y. & Chateauneuf, A. 2010. Benchmark study for reliability-based design optimization. Structural and Multidisciplinary Optimization 41: 277–294.
Casciati, S. 2007. Including structural robustness in reliability-oriented optimal design. In George Deoda-tis and Pol D. Spanos (eds.), Computational Stochastic Mechanics; Proc. CSM-5 Fifth International Confer-ence, Rhodes, 21–23 June, 2006. Rotterdam: Millpress.
157–161.
Casciati, S. 2008. Stiffness identification and damage localization via differential evolution algorithms. Jour-nal of Structural Control and Health Monitoring 15(3):
436–449.
Casciati, S. & Faravelli, L. 2008. Building a Robustness Index. In Michael Faber (ed.): Action TU0601 Robust-ness of Structures; Proceedings of the 1st Workshop, ETH Zurich, 4–5 February 2008. Zurich: Reprozent-rale ETH Hönggerberg, 49–56.
Faber, M.H. & Stewart, M.G. 2003. Risk assessment for civil engineering facilities: critical overview and dis-cussion. Reliability Engineering and System Safety 80:
173–184.
Rackwitz, R. 2002. Optimization and risk acceptability based on the life quality index. Structural Safety 24:
297–331.
Storn, R. & Price, K. 1997. Differential evolution—a simple and efficient heuristic for global optimization over continuous spaces. Journal of Global Optimiza-tion 11: 341–359.